It doesn't need to be either/or. Also most people check their parse at the end of an encounter or after a wipe, sometimes they'll glance over to see where they're at but no one's eyes are glued to their parser. Everyone that wants their number to go higher knows that you do 0 damage when you're dead, I think you're being a bit hyperbolic here.
Also you don't NEED a parser to do better - there are many aspects you can improve on that a parser won't help you with. However, if you want to improve your DPS contribution then a parser is a very useful tool for doing so as it lets you know where you are and (after adjustments) how much you've improved.